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Spectral gaps in Wasserstein distances and the 2D stochastic NavierStokes equations
, 2006
"... We develop a general method that allows to show the existence of spectral gaps for Markov semigroups on Banach spaces. Unlike most previous work, the type of norm we consider for this analysis is neither a weighted supremum norm nor an Ł ptype norm, but involves the derivative of the observable as ..."
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Cited by 15 (7 self)
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We develop a general method that allows to show the existence of spectral gaps for Markov semigroups on Banach spaces. Unlike most previous work, the type of norm we consider for this analysis is neither a weighted supremum norm nor an Ł ptype norm, but involves the derivative of the observable as well and hence can be seen as a type of 1–Wasserstein distance. This turns out to be a suitable approach for infinitedimensional spaces where the usual Harris or Doeblin conditions, which are geared to total variation convergence, regularly fail to hold. In the first part of this paper, we consider semigroups that have uniform behaviour which one can view as an extension of Doeblin’s condition. We then proceed to study situations where the behaviour is not so uniform, but the system has a suitable Lyapunov structure, leading to a type of Harris condition. We finally show that the latter condition is satisfied by the twodimensional stochastic NavierStokers equations, even in situations where the forcing is extremely degenerate. Using the convergence result, we show shat the stochastic NavierStokes equations ’ invariant measures depend continuously on the viscosity and the structure of the forcing. 1
Maslowski B.: Lower estimates of transition densities and bounds on exponential ergodicity for stochastic PDE’s
"... A formula for the transition density of a Markov process defined by an infinitedimensional stochastic equation is given in terms of the Ornstein–Uhlenbeck bridge and a useful lower estimate on the density is provided. As a consequence, uniform exponential ergodicity and Vergodicity are proved for ..."
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Cited by 5 (2 self)
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A formula for the transition density of a Markov process defined by an infinitedimensional stochastic equation is given in terms of the Ornstein–Uhlenbeck bridge and a useful lower estimate on the density is provided. As a consequence, uniform exponential ergodicity and Vergodicity are proved for a large class of equations. We also provide computable bounds on the convergence rates and the spectral gap for the Markov semigroups defined by the equations. The bounds turn out to be uniform with respect to a large family of nonlinear drift coefficients. Examples of finitedimensional stochastic equations and semilinear parabolic equations are given. 1. Introduction. The
ANALYSIS OF EQUILIBRIUM STATES OF MARKOV SOLUTIONS TO THE 3D NAVIERSTOKES EQUATIONS DRIVEN BY ADDITIVE NOISE
, 709
"... ABSTRACT. We prove that every Markov solution to the three dimensional NavierStokes equation with periodic boundary conditions driven by additive Gaussian noise is uniquely ergodic. The convergence to the (unique) invariant measure is exponentially fast. Moreover, we give a wellposedness criterion ..."
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Cited by 4 (3 self)
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ABSTRACT. We prove that every Markov solution to the three dimensional NavierStokes equation with periodic boundary conditions driven by additive Gaussian noise is uniquely ergodic. The convergence to the (unique) invariant measure is exponentially fast. Moreover, we give a wellposedness criterion for the equations in terms of invariant measures. We also analyse the energy balance and identify the term which ensures equality in the balance. 1.
Approximations to the Stochastic Burgers equation
, 2010
"... This article is devoted to the numerical study of various finite difference approximations to the stochastic Burgers equation. Of particular interest in the onedimensional case is the situation where the driving noise is white both in space and in time. We demonstrate that in this case, different f ..."
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This article is devoted to the numerical study of various finite difference approximations to the stochastic Burgers equation. Of particular interest in the onedimensional case is the situation where the driving noise is white both in space and in time. We demonstrate that in this case, different finite difference schemes converge to different limiting processes as the mesh size tends to zero. A theoretical explanation of this phenomenon is given and we formulate a number of conjectures for more general classes of equations, supported by numerical evidence. 1